A framework of Rogers-Ramanujan identities and their arithmetic properties

The two Rogers-Ramanujan $q$-series $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n(n+\sigma )}}{(1-q)\cdots (1-q^n)},$$ where $\sigma =0,1$, play many roles in mathematics and physics. By the Rogers-Ramanujan identities, they are essentially modular functions. Their quotient, the Rogers-Ramanujan continued fraction, has the special property that its singular values are algebraic integral units. We find a framework which extends the Rogers-Ramanujan identities to doubly-infinite families of $q$-series identities. If $a\in \{1,2\}$ and $m,n\ge 1$, then we have $$\sum _{\begin{array}{c}\lambda {\lambda }_1\le m\end{array}}{q}^{a\mathrm{\mid }\lambda \mathrm{\mid }}{P}_{2\lambda }(1,q,q^2,\dots ;q^n)=\text{"infinite product modular function"},$$ where the $P_{\lambda }(x_1,x_2,\dots ;q)$ are Hall-Littlewood polynomials. These $q$-series are specialized characters of affine Kac–Moody algebras. Generalizing the Rogers-Ramanujan continued fraction, we prove in the case of ${\text{A}}_{2n}^{(2)}$ that the relevant $q$-series quotients are integral units.